We deduce that \(X_T^c\)  converges to \(Y\)  in \(L^{r_1^*}\)  as \(c\)  goes to \(+\infty \)  and thus in distribution and, since \(U\)  is continuous, \( \lim _{c \rightarrow + \infty } U(\mathcal {L}(X_T^c)) = U(Y) < 0\) . From a mimicking argument of [1]}, Corollary 3.7, there exist two measurable functions \(\tilde{b} :[0,T] \times \mathbb {R}^d \rightarrow \mathbb {R}^d \)  and \(\tilde{\sigma } : [0,T] \times \mathbb {R}^d \rightarrow \mathbb {M}_d(\mathbb {R})\) , such that for almost all \(t\)  in \([0,T] \)  (for the Lebesgue measure),
\(\left\lbrace \begin{array}{ll}\tilde{b}(t,X^c_t) = \mathbb {E}( c(Y-X^c_t) |X^c_t ) \\\tilde{\sigma } ^t \tilde{\sigma }(t,X_t^c) = \mathbb {E}( \sigma ^t\sigma (X_t^c,a^2) |X_t^c) =\sigma ^t\sigma (X_t^c,a^2) .\end{array}\right.\) 
